That's what the duc de Soubise, a major leader of the Huguenots, did in 1625, capturing six French naval ships and escaping with them to La Rochelle.
Bureaucrats in Early Modern France could be expected to act as policemen on the spot if they saw a crime in progress. Diplomacy and espionage was carried out by marquises, dukes, and princes. Presidents (judges) both started and suppressed riots.
Again, I think it's a mistake to superimpose our modern understanding of leadership and administration on earlier time periods. And in a fantasy setting, you have even more leeway to use history as inspiration and license.
I'm not familiar with the duke you cite but I know my way around Western Renaissance and earlier periods. I suspect that if the raid you mentioned captured 6 ships, his duties might still have been more directing than swinging a cutlass for most of the action. Directing the number of sailors necessary to capture 6 ships does not strike me as much in the way of typical D&D adventuring where the focus is on the success and failure of a small group's actions.
I think there are plenty of examples of lower level leaders acting in ways consistent with what seems to occur in most D&D games. But by the time you get to senior leaders, from historical precedent, it seems to me it gets pretty hard to keep the leader in the thick of things for the repeated actions necessary to carry a campaign.
Worse, it can also get very focused on one PC. Maybe you can contrive a number of actions around the King or Admiral or what not but the other players are playing second fiddle to the PC who is the leader, generally not an ideal situation.
The historical precedents I can think of, akin to your duc, are fairly limited and even for the person in question, small in number. There are examples like Titus who as a Roman general, during the campaign against the Jewish Rebellion 66-70 CE, several times went off on his own (with a very small group) for actions that fit a more classic D&D-encounter but this was 3-4 times over the course of 4 years. There have been rulers at times who liked to wade into the thick of things in army battles but that doesn't strike me as the type of D&D encounter that is all that interesting or meaningful (the leader rallied his troops by his action, the leader lost the battle by being unable to direct his troops, that's about it. The number of people the leader and his bodyguard actually killed was irrelevant.)
The Romans had a special recognition for generals who killed an opposing general and once Rome's armies grew large, this just didn't happen, despite the fact that Roman generals burned with ambition and would have loved to achieve it. I'm a little hazy on the details but if memory serves, there was a general who achieved this at the end of the Roman Republic and this event was remarkable because it hadn't happened in hundreds of years. IIRC he was denied the honors for this achievement by Octavian(?) so as not to outshine him.
Certainly there have been rulers who started as leaders of small bands and did some very daring and D&D-ish things to earn great renown and title. But once they had achieved great renown and the ability to deploy larger forces, it seems to me the scope for the leader historically to go off and act in a way where his sword arm was significant was very limited and in most D&D games, similarly ought to be limited.
The King of France wouldn't slink off from France with 4 good knights and take a citadel in the holy land by storm. He went to the Holy Land with a large army and remained in the midst of that army.
You can do what you want in your own campaign of course. The lower ranked the leader, the less the issue and the mayor of the OP is pretty low ranked it seems. But for someone to knock a player who thinks the demands of leadership are incompatible with adventuring seems unreasonable to me. If he thinks that his character is no longer able to adventure, he should be rewarded for playing his character well by being allowed to ease into a new character with little penalty.